Eric Allender, Robert Beals, Mitsunori Ogihara

We characterize the complexity of some natural and important

problems in linear algebra. In particular, we identify natural

complexity classes for which the problems of (a) determining if a

system of linear equations is feasible and (b) computing the rank of

an integer matrix, ...
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Alina Beygelzimer, Mitsunori Ogihara

We investigate the complexity of enumerative approximation of

two elementary problems in linear algebra, computing the rank

and the determinant of a matrix. In particular, we show that

if there exists an enumerator that, given a matrix, outputs a

list of constantly many numbers, one of which is guaranteed to

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Agostino Capponi

Communication complexity is concerned with the question: how much information do the participants of a communication system need to exchange in order to perform certain tasks? The minimum number of bits that must be communicated is the deterministic communication complexity of $f$. This complexity measure was introduced by Yao \cite{1} ... more >>>

Nitin Saxena, C. Seshadhri

We show that the rank of a depth-3 circuit (over any field) that is simple,

minimal and zero is at most O(k^3\log d). The previous best rank bound known was

2^{O(k^2)}(\log d)^{k-2} by Dvir and Shpilka (STOC 2005).

This almost resolves the rank question first posed by ...
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Emanuele Viola, Emanuele Viola

We prove that to store n bits x so that each

prefix-sum query Sum(i) := sum_{k < i} x_k can be answered

by non-adaptively probing q cells of log n bits, one needs

memory > n + n/log^{O(q)} n.

Our bound matches a recent upper bound of n +

n/log^{Omega(q)} ...
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Massimo Lauria

Ramsey Theorem is a cornerstone of combinatorics and logic. In its

simplest formulation it says that there is a function $r$ such that

any simple graph with $r(k,s)$ vertices contains either a clique of

size $k$ or an independent set of size $s$. We study the complexity

of proving upper ...
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Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar Lovett

Let $\mathbb{F} = \mathbb{F}_p$ for any fixed prime $p \geq 2$. An affine-invariant property is a property of functions on $\mathbb{F}^n$ that is closed under taking affine transformations of the domain. We prove that all affine-invariant property having local characterizations are testable. In fact, we show a proximity-oblivious test for ... more >>>

Nitin Saxena

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.

more >>>Albert Atserias, Massimo Lauria, Jakob NordstrÃ¶m

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n^Omega(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size n^O(w) is essentially tight. ... more >>>

Massimo Lauria, Jakob NordstrÃ¶m

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size n^{Omega(d)} for values of d = d(n) from constant all the way up to n^{delta} for some universal constant delta. This shows that ... more >>>